[Paper Review] A cure for instabilities due to advection-dominance in POD solution to advection-diffusion-reaction equations

Mejdi Azaïez, Tomás Chacón Rebollo+1 more|arXiv (Cornell University)|Jul 12, 2019

Model Reduction and Neural Networks58 references18 citations

TL;DR

This paper introduces a stabilizing post-processing (SPP) strategy to address numerical instabilities in Proper Orthogonal Decomposition Reduced Order Models (POD-ROM) for advection-diffusion-reaction equations in the strongly advection-dominated regime. By applying SPP during both offline snapshot generation and online solution computation, the method significantly reduces spurious oscillations and improves accuracy, achieving near-offline-level performance even with low-mode POD bases.

ABSTRACT

In this paper, we propose to improve the stabilized POD-ROM introduced by S. Rubino in [37] to deal with the numerical simulation of advection-dominated advection-diffusion-reaction equations. In particular, we introduce a stabilizing post-processing strategy that will be very useful when considering very low diffusion coefficients, i.e. in the strongly advection-dominated regime. This strategy is applied both for the offline phase, to produce the snapshots, and the reduced order method to simulate the new solutions. The new process of a posteriori stabilization is detailed in a general framework and applied to advection-diffusion-reaction problems. Numerical studies are performed to discuss the accuracy and performance of the new method in handling strongly advection-dominated cases.

Motivation & Objective

• Address the well-known instability of standard POD-Galerkin ROMs in convection-dominated flows due to insufficient dissipation of unresolved small-scale modes.
• Overcome the limitations of existing stabilized POD-ROMs like SD-POD-ROM when diffusion coefficients are very low, i.e., in the strongly advection-dominated regime.
• Develop a general-purpose a posteriori stabilization framework applicable to both offline snapshot generation and online ROM solution computation.
• Improve the accuracy and robustness of reduced-order models in long-time simulations and problems with sharp layers or discontinuities.
• Demonstrate that post-processing can effectively suppress noisy mode influence without requiring additional online computational cost beyond standard ROM solve.

Proposed method

• Introduce a stabilizing post-processing (SPP) technique applied to both the snapshot snapshots in the offline phase and the online ROM solution.
• Apply the SPP as a filtering operation that dampens high-frequency oscillations in the reduced solution, mimicking numerical stabilization.
• Integrate the SPP within the standard POD-ROM framework, using a stabilized finite element formulation (SD-POD-ROM) as the baseline.
• Use a filtered advection stabilization approach in the offline phase to generate less noisy snapshots, reducing the impact of spurious modes in the reduced basis.
• Apply the same SPP technique in the online phase to stabilize the reduced solution, particularly near sharp layers or fronts.
• Formulate the method in a general framework applicable to advection-diffusion-reaction problems, enabling extension to other PDEs.

Experimental results

Research questions

• RQ1 Can a post-processing strategy effectively stabilize POD-ROM solutions in the strongly advection-dominated regime where standard and stabilized ROMs fail?
• RQ2 Does applying post-processing during the offline phase improve the quality of snapshots and reduce the influence of noisy modes in the reduced basis?
• RQ3 How does the combination of SPP in both offline and online stages affect the accuracy and oscillation levels in reduced-order solutions?
• RQ4 Can the proposed method achieve accuracy comparable to full-order models while maintaining low computational cost?
• RQ5 What is the performance of the method in long-time simulations and problems with sharp moving fronts?

Key findings

• For ν = 10⁻⁸ and r = 90, the SD-ROM with online SPP achieved an L² error of 0.0589, approaching the offline solution accuracy.
• The SPP method reduced oscillations significantly: at r = 90, the SD-ROM with post-processing had lower error (0.0589) than the standard G-ROM (0.1224) and even the G-ROM with post-processing (0.0884).
• Even with only r = 30 modes (capturing 99.71% of kinetic energy), the SPP-enhanced SD-ROM reduced oscillations and improved stability compared to standard ROMs.
• The method effectively localized and suppressed oscillations near sharp layers, while preserving the front amplitude better than standard or G-ROM approaches.
• The SPP strategy proved effective across different numbers of POD modes, with oscillations diminishing as r increased, indicating robustness and scalability.
• Numerical results show that SPP can be a highly effective, low-cost enhancement to existing stabilized POD-ROMs, especially in convection-dominated settings.